$A$ mixture of $2$ moles of helium gas (atomic mass $= 4$ $amu$) and $1$ mole of argon gas (atomic mass $= 40$ $amu$) is kept at $300$ $K$ in a container. The ratio of the rms speeds $(\frac{V_{rms}(helium)}{V_{rms}(argon)})$ is

In a metre bridge experiment,the ratio of the left gap resistance to the right gap resistance is $2:3$. The balance point from the left is: (in $~cm$)

$1+\frac{1}{4}+\frac{1 \cdot 3}{4 \cdot 8}+\frac{1 \cdot 3 \cdot 5}{4 \cdot 8 \cdot 12}+\ldots$ is equal to

If $\sqrt{x} + \frac{1}{\sqrt{x}} = 2\cos \theta$,then ${x^6} + {x^{-6}} = $

An electron moves with speed $2 \times 10^5 \ m/s$ along the positive $x$-direction in the presence of a magnetic induction $\vec{B} = \hat{i} + 4\hat{j} - 3\hat{k}$ (in $Tesla$). The magnitude of the force experienced by the electron in Newtons is (charge on the electron = $1.6 \times 10^{-19} \ C$).

The angle between the lines $\vec{r}=(2 \hat{i}-3 \hat{j}+\hat{k})+\lambda(\hat{i}+4 \hat{j}+3 \hat{k})$ and $\vec{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(\hat{i}+2 \hat{j}-3 \hat{k})$ is